Editing WKB approximation (section)

=== Behavior near the turning points ===
We now consider the behavior of the wave function near the turning points. For this, we need a different method. Near the first turning points, {{math|''x''<sub>1</sub>}}, the term <math>\frac{2m}{\hbar^2}\left(V(x)-E\right)</math> can be expanded in a power series,
<math display="block">\frac{2m}{\hbar^2}\left(V(x)-E\right) = U_1 \cdot (x - x_1) + U_2 \cdot (x - x_1)^2 + \cdots\;.</math>

To first order, one finds
<math display="block">\frac{d^2}{dx^2} \Psi(x) = U_1 \cdot (x - x_1) \cdot \Psi(x).</math>
This differential equation is known as the [[Airy equation]], and the solution may be written in terms of [[Airy function]]s,<ref>{{harvnb|Hall|2013}} Section 15.5</ref>
<math display="block">\Psi(x) = C_A \operatorname{Ai}\left( \sqrt[3]{U_1} \cdot (x - x_1) \right) + C_B \operatorname{Bi}\left( \sqrt[3]{U_1} \cdot (x - x_1) \right)= C_A \operatorname{Ai}\left( u \right) + C_B \operatorname{Bi}\left( u \right).</math>

Although for any fixed value of <math>\hbar</math>, the wave function is bounded near the turning points, the wave function will be peaked there, as can be seen in the images above. As <math>\hbar</math> gets smaller, the height of the wave function at the turning points grows. It also follows from this approximation that:

<math display="block">\frac{1}{\hbar}\int p(x) dx = \sqrt{U_1} \int \sqrt{x-a}\, dx = \frac 2 3 (\sqrt[3]{U_1} (x-a))^{\frac 3 2} = \frac 2 3 u^{\frac 3 2}</math>